Stability analysis of inhomogeneous equilibrium for axially and transversely excited nonlinear beam

Emine Kaya; Eugenio Aulisa; Akif Ibragimov; Padmanabhan Seshaiyer

doi:10.3934/cpaa.2011.10.1447

Article Contents

2011, Volume 10, Issue 5: 1447-1462. Doi: 10.3934/cpaa.2011.10.1447

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Stability analysis of inhomogeneous equilibrium for axially and transversely excited nonlinear beam

1.
Department of Mathematics and Statistics, Texas Tech University, Lubbock TX, 79409-1042
2.
Department of Mathematical Sciences, George Mason University, Fairfax VA, 22030

Received: April 30, 2009

Revised: October 31, 2010

Published: August 2011

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Abstract

In this work we consider the dynamical response of a non-linear beam with viscous damping, perturbed in both the transverse and axial directions. The system is modeled using coupled non-linear momentum equations for the axial and transverse displacements. In particular we show that for a class of boundary conditions (beam clamped at the extremes) and uniformly distributed load, there exists a non-uniform equilibrium state. Different models of damping are considered: first, third and fifth order dissipation terms. We show that in all cases in the presence of the damping forces, the excited beam is stable near the equilibrium for any perturbation. An energy estimate approach is used in order to identify the space in which the solution of the perturbed system is stable.

Keywords:

Stability analysis,
non-linear partial differential equations,
Euler-Bernoulli beam.

Mathematics Subject Classification: Primary: 49K20, 70K20; Secondary: 74K10.

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